L(s) = 1 | + 9-s − 25-s + 2·29-s + 2·37-s − 2·53-s + 81-s + 2·109-s + 2·113-s + ⋯ |
L(s) = 1 | + 9-s − 25-s + 2·29-s + 2·37-s − 2·53-s + 81-s + 2·109-s + 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.378070050\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.378070050\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.869964843523226377236808894837, −8.003839731444062893456838383234, −7.48304811721430448069905929897, −6.54078160069594979143883411571, −6.01138669903185719912772568633, −4.82014995632150533511233271147, −4.33062359944352431845235746335, −3.30904663177662742756268046106, −2.26918576342345652086512586798, −1.13088579090641869150858575814,
1.13088579090641869150858575814, 2.26918576342345652086512586798, 3.30904663177662742756268046106, 4.33062359944352431845235746335, 4.82014995632150533511233271147, 6.01138669903185719912772568633, 6.54078160069594979143883411571, 7.48304811721430448069905929897, 8.003839731444062893456838383234, 8.869964843523226377236808894837